Higman's conjecture on the number of conjugacy classes of $$U(n,q)$$

Igor Pak (UCLA)

In 1960, Higman asked whether the number $$f(n,q)$$ of conjugacy classes of $$n\times n$$ unitriangular matrices $$U(n,q)$$ over the finite field with $$q$$ elements, is polynomial in $$q$$ for every $$n$$. I will survey what is known about this problem and explain the connections to enumerative combinatorics and group representation theory. I will then describe our recent efforts to prove the conjecture for small $$n$$ and disprove it for large $$n$$. (Joint work with Andrew Soffer.)